Photonic crystals in biology - NanoTR-VI

PP andPoster Session, Thursday, June 17Theme F686 - N1123Analytical Solution of an Electrokinetic Flow in a Nano-Channel using Curvilinear Coordinates111UMehdi MostofiUP P*, Davood D. GanjiP Mofid Gorji-BandpyP1PDepartment of Mechanical Engineering, Noshiravani University of Technology, Babol, IranAbstract-In this paper, an electrokinetic flow of an electrolyte in a 15 nm radius nano-channel will be studied. This study will be with existenceof the Electric Double Layer (EDL) and fully analytical. Governing equations for electrokinetic phenomena are Poisson-Boltzmann, Navier-Stokes, species and mass conservation equations. Induced electric potential force the electrolyte ions and decrease the mass flow rate. In thispaper, it is assumed that, zeta potential has small quantity. In this paper, after getting the equations set from the literature and transforming itinto curvilinear coordinates, the set will be simplified and be solved analytically for small zeta potentials in a nano-channel.One of the most important subsystems of the micro- andnano- fluidic devices is their passage or “Micro- and Nano-Channel”. Nano-channel term is referred to channels withhydraulic diameter less than 100 nanometers [1]. By decreasein size and hydraulic diameter some of the physical parameterssuch as surface tension will be more significant while they arenegligible in normal sizes.Concentrating surface loads in liquid – solid interface makesthe EDL to be existed. If the loads are concentrated in the endof nano-channels, a potential difference will be generated thatforces the ions in the nano-channel. However, induced electricfield is discharged by electric conduction of the electrolyte.Rice and Whitehead [2], Lu and Chan [3] and Ke and Liu[4] studied the flow in capillary tube. None of them solved theproblem based on the curvilinear coordinates system. Also, allof them studied the problem with existence of the pressuregradient while in the modern applications, the pressuregradient can be eliminated and consequently, solving theproblem considering this fact is necessary. In this paper, forsmall zeta potentials without pressure gradient will be studiedbased on the curvilinear coordinates in a capillary tube.In electrokinetic processes, for the most general form of thestudy, seven nonlinear equations govern an electrokineticprocess [5]. In this paper, by some simplifications that will bementioned later, this set will be made simpler.Next in this work, a very long nano- tube will beinvestigated. According to the fact that reference length of thetube according to x direction (L) is very larger than capillaryradius (R) and reference amount for theta ( ), we can neglectseveral terms of the equations. In addition, it is assumed that,electric potential in the x direction is constant. According tothese assumptions, the equations mentioned below will beavailable:1 r X p X m2(1)r r r 1r urr r 1 Xrr r r1 Xrr r rpm eE0RT2 F U0Xp Xm(2) Xp 0r(3) X m0r(4)By applying boundary conditions(no slip condition at walland free stream velocity in center of nano-channel for velocityfield and zeta potential at wall and finite amount of it at centerof the nano-channel for potential field), we have the followingfigures. Figures (a) and (b) show the results for velocity andpotential fields respectively.In summary, by considering curvilinear coordinates andusing Taylor series, some derivation of Developed BesselODE has been derived and solved for Poisson-Boltzmannequation. In addition, velocity profile in nano-tube has beenachieved for small amounts of zeta potentials. Results thoseare derived by curvilinear coordinates are in good agreementwith those of resulted by rectilinear ones in [5].Figure 1. Normalized distribution of potential as a function ofnormalized radius.Figure 2. Normalized velocity profile as a function ofnormalized radius.* Corresponding author: HTmehdi_mostofi@yahoo.comT[1] S. Kandlikar, et. al, Heat Transfer and Fluid Flow inMinichannels and Microchannels. Elsevier Limited, Oxford (2006).[2] Rice, C.L. and Whitehead, R. J. Phys. Chem., 69(11), 4017–4023(1965)[3] W.Y. Lo, and K. Chan. J. Chem. Phys., 143, 339–353 (1994)[4] H. Keh, and Y.C. Liu, J. Colloids and Interface Surfaces, 172,222–229 (1995)[5] Zheng, Z.: Electrokinetic Flow in Micro- and Nano- FluidicComponents. Ohio State University, (2003).6th Nanoscience and Nanotechnology Conference, zmir, 2010 681